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Real Mathematical Analysis
Based on an honors course taught by the author at UC Berkeley, this introduction to undergraduate real analysis gives a different emphasis by stressing the importance of pictures and hard problems. Topics a natural construction of the real numbers, four-dimensional visualization, basic point-set topology, function spaces, multivariable calculus via differential forms (leading to a simple proof of the Brouwer Fixed Point Theorem), and a pictorial treatment of Lebesgue theory. Over 150 detailed illustrations elucidate abstract concepts and salient points in proofs. The exposition is informal and relaxed, with many helpful asides, examples, some jokes, and occasional comments from mathematicians, such as Littlewood, Dieudonné, and Osserman. This book thus succeeds in being more comprehensive, more comprehensible, and more enjoyable, than standard introductions to analysis.
New to the second edition of Real Mathematical Analysis is a presentation of Lebesgue integration done almost entirely using the undergraph approach of Burkill. Payoffs concise picture proofs of the Monotone and Dominated Convergence Theorems, a one-line/one-picture proof of Fubini's theorem from Cavalieri’s Principle, and, in many cases, the ability to see an integral result from measure theory. The presentation includes Vitali’s Covering Lemma, density points ― which are rarely treated in books at this level ― and the almost everywhere differentiability of monotone functions. Several new exercises now join a collection of over 500 exercises that pose interesting challenges and introduce special topics to the student keen on mastering this beautiful subject.
New to the second edition of Real Mathematical Analysis is a presentation of Lebesgue integration done almost entirely using the undergraph approach of Burkill. Payoffs concise picture proofs of the Monotone and Dominated Convergence Theorems, a one-line/one-picture proof of Fubini's theorem from Cavalieri’s Principle, and, in many cases, the ability to see an integral result from measure theory. The presentation includes Vitali’s Covering Lemma, density points ― which are rarely treated in books at this level ― and the almost everywhere differentiability of monotone functions. Several new exercises now join a collection of over 500 exercises that pose interesting challenges and introduce special topics to the student keen on mastering this beautiful subject.
489 pages, Hardcover
First published March 1, 2002
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260 people want to read
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Displaying 1 - 14 of 14 reviews
April 20, 2007
prof wrote this book. offered 5 dollars for each mathematical mistake, 1 dollar for a typo. i found a 2 dollar mistake: a minor mathematical mistake. hard class; honor's version of analysis. half of the class died of headache, half of the rest died of heartache, half of the rest died of loss of self-esteem...eventually we'll all die.
November 30, 2015
The book starts studying Numbers, Actually real numbers and constructs them from rational numbers with Dedekind Cuts, Which we expect to see these parts in a Set theory book or something about Math bases. This chapter is well written, But not what we think of the word "Analysis".
Second chapter is about Metric Spaces, I think it's the best chapter of this book,Very clear and educational, not theorem_Lemma style, it seriously tries to "teach" you the concepts, This chapter_As the name tells you : A taste of topology_ doesn't have very analytic taste again. But it's a very good start to understanding Topology even though Maybe it does not teach you topology directly.
Rest of the book has more Analytic Topics, Like Derivative, Integral, Function Spaces. These chapters are more familiar and somehow you can see them as applications of chapter 2.
The book is not Rich enough on multivariable part. Maybe it's better to look at 'Calculus on Manifolds' by Spivak for this topic.
I repeat again, the book really cares about how you will learn concepts, it is full of figures, examples, questions. Very good book to read in two semesters for an undergraduate student.
Second chapter is about Metric Spaces, I think it's the best chapter of this book,Very clear and educational, not theorem_Lemma style, it seriously tries to "teach" you the concepts, This chapter_As the name tells you : A taste of topology_ doesn't have very analytic taste again. But it's a very good start to understanding Topology even though Maybe it does not teach you topology directly.
Rest of the book has more Analytic Topics, Like Derivative, Integral, Function Spaces. These chapters are more familiar and somehow you can see them as applications of chapter 2.
The book is not Rich enough on multivariable part. Maybe it's better to look at 'Calculus on Manifolds' by Spivak for this topic.
I repeat again, the book really cares about how you will learn concepts, it is full of figures, examples, questions. Very good book to read in two semesters for an undergraduate student.
May 16, 2020
yes, this is a literary review of a math textbook. i would argue that actually such reviews are as necessary as reviews of already-shortlisted novels and essays, given the vast amounts of mind-numbingly jargon-filled passive voice crowding today's market of anything even slightly "academic" blocking non-phd students from fully appreciating the beauty of often esoterically written concepts.
reading this felt like having a very strange, philosophical yet somehow technical conversation with a witty close friend. for anyone who is dipping their toes into real analysis for the first time, i highly recommend pugh! his writing style is simply so down-to-earth and endlessly entertaining, despite the abstract nature of the topic discussed. look out for the pictorial descriptions of seemingly dense concepts, and the breaths of fresh air that he provides every so often in the form of mathematical poetry! not to mention, he accomplishes all of this while maintaining a level of mathematical rigor expected of university-level textbook writers. truly a rare gift.
reading this felt like having a very strange, philosophical yet somehow technical conversation with a witty close friend. for anyone who is dipping their toes into real analysis for the first time, i highly recommend pugh! his writing style is simply so down-to-earth and endlessly entertaining, despite the abstract nature of the topic discussed. look out for the pictorial descriptions of seemingly dense concepts, and the breaths of fresh air that he provides every so often in the form of mathematical poetry! not to mention, he accomplishes all of this while maintaining a level of mathematical rigor expected of university-level textbook writers. truly a rare gift.
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August 28, 2014Easy to understand.
June 30, 2018
One of the very enjoyable books on mathematics I have read. The topics are simplified and detailed for an undergraduate level study. The reader is given plenty of visualisations and exercises to understand the concepts ranging from lagrangian multipliers to measure theory. It may serve as a great primer for anybody who is interested in real analysis but has shied away from a heavier algebraic treatment.
May 1, 2024
This is a great book. Good examples. Good exercises.
Ideas are presented in a way that’s as easy to understand and possible.
At the end of the day Analysis is hard asf and you are going to want to die but this book makes it possible.
Ideas are presented in a way that’s as easy to understand and possible.
At the end of the day Analysis is hard asf and you are going to want to die but this book makes it possible.
April 17, 2025
Good content, useful for almost any undergrad analysis course sans measure theory. The style is similar to listening to a lecture's transcription and for some can be difficult to follow at times. Interesting problems.
June 6, 2017
an intuitionistic approach to mathematical analysis. The construction of real numbers via Dedekind cut is fun to read. Suitable as a supplement to Baby Rudin.
August 30, 2017
Fewer proofs, more examples, please!!
October 17, 2020
Test 1 average was <30%
November 22, 2025
honestly if not for the pretty unacceptable errata would probably recommend as the replacement for rudin which i think categorically refuses to capture that analysis can be beautiful
April 11, 2016
This is a terrific book. Its explanations are crystal clear and concise, and its exercises are really interesting. Unlike many other math books, this is definitely a book that is written to be read, and the book is written in a wonderful conversational style.
The chapter on metric space theory is absolutely incredible, and the construction of the real numbers in chapter one is really compelling. While the book can be light on examples, the author always provides enough of them to motivate the more rigorous definitions and proofs.
This books greatest strength is does more than just teaches real analysis--it teaches how to think and solve problems in analysis. The author peppers the text with his advice for solving problems or conceptualizing certain concepts, and it really helps the book.
Could not recommend this book more.
The chapter on metric space theory is absolutely incredible, and the construction of the real numbers in chapter one is really compelling. While the book can be light on examples, the author always provides enough of them to motivate the more rigorous definitions and proofs.
This books greatest strength is does more than just teaches real analysis--it teaches how to think and solve problems in analysis. The author peppers the text with his advice for solving problems or conceptualizing certain concepts, and it really helps the book.
Could not recommend this book more.
April 26, 2016
Undoubtedly, this is a heavy book. I'm lucky to have a wonderful professor who carefully guides us through the material and often explains things from a great, different point of view.
If you're one of those students who try to do all problems in the section, good luck! There are over 100 problems in each chapter, not including the UC Berkeley qualifying exam problems (which I think is really cool). Many problems are challenging, but then again, don't you love it when you figure one of those out?
Nevertheless, the book is a great studying source, whether you want to apply the theorems to your problems, or learn the proofs in great length.
If you're one of those students who try to do all problems in the section, good luck! There are over 100 problems in each chapter, not including the UC Berkeley qualifying exam problems (which I think is really cool). Many problems are challenging, but then again, don't you love it when you figure one of those out?
Nevertheless, the book is a great studying source, whether you want to apply the theorems to your problems, or learn the proofs in great length.
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October 31, 2014aa
Displaying 1 - 14 of 14 reviews